Section 6.3Exercises

347

CLARIFYING THE CONCEPTS

Question 6.183

1. Describe when the Poisson probability distribution is used. (p. 341)

6.3.1

When observing the number of occurrences of an event within a fixed interval of space and time

Question 6.184

2. For a Poisson random variable , to what does refer? (p. 341)

Question 6.185

3. Explain the requirements for the Poisson probability distribution. (p. 341)

6.3.3

The occurrences must be random, independent, and uniformly distributed over the given interval.

Question 6.186

4. What are the conditions for using the Poisson distribution to approximate the binomial distribution? (p. 345)

PRACTICING THE TECHNIQUES

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To do Check out Topic
Exercises 5–8 Example 21 Recognizing when to use
the Poisson distribution
Exercises 9–14 Example 22 Finding probabilities
using the Poisson
distribution
Exercises 15–18 Example 23 Applying the mean and
standard deviation
Exercises 19–22 Example 24 Poisson approximation to
the binomial distribution

For each of the following situations, state whether or not the random variable follows a Poisson probability distribution. If not, state why not.

Question 6.187

5. is the number of sheep in an area of land that is part pasture and part forest.

6.3.5

Does not follow a Poisson distribution

Question 6.188

6. is the number of fraternity members on a certain campus who will consume alcoholic beverages this weekend.

Question 6.189

7. is the number of vehicles to pass by a certain point on a rural stretch of road between 11 P.M. and midnight.

6.3.7

Follows a Poisson distribution

Question 6.190

8. is the number of soldiers marching in a one-mile-long parade.

For each of the following situations, assume that the requirements for the Poisson probability distribution are met. Use the given mean to find the indicated probability.

Question 6.191

9. ; find the probability that equals 8.

6.3.9

0.1126

Question 6.192

10. ; find the probability that equals 9.

Question 6.193

11. ; find the probability that is at most 3.

6.3.11

0.6472

Question 6.194

12. ; find the probability that is greater than 3. (Hint: First find .)

Question 6.195

13. ; find the probability that is at most 2.

6.3.13

0.2381

Question 6.196

14. ; find the probability that is greater than 2. (Hint: First find .)

For each of the following situations, assume that the requirements for the Poisson probability distribution are met and do the following:

  1. Calculate the standard deviation.
  2. Find the values of that would be considered moderately unusual.

Question 6.197

15.

6.3.15

(a) 4.4721 (b) 11.0558, 28.9442, any values less than or equal to 11 or greater than or equal to 29 will be considered moderately unusual

Question 6.198

16.

Question 6.199

17.

6.3.17

(a) 3.1623 (b) 3.6754, 16.3246, any values less than or equal to 3 or greater than or equal to 17 will be considered moderately unusual

Question 6.200

18.

For each of the following situations, do the following:

  1. Verify that the requirements are met for using the Poisson approximation for the binomial distribution.
  2. Use the Poisson approximation for the binomial distribution to approximate the indicated probability.

Question 6.201

19. , . Approximate the probability that .

6.3.19

(a) and (b) 0.0948

Question 6.202

20. , . Approximate the probability that .

Question 6.203

21. , . Approximate the probability that .

6.3.21

(a) and (b) 0.0993

Question 6.204

22. , . Approximate the probability that .

APPLYING THE CONCEPTS

Assume that the requirements for the Poisson probability distribution are met for Exercises 23–31. Also, when asked to approximate a probability, use the Poisson approximation for the binomial distribution.

Question 6.205

23. Trees per Acre. The mean number of trees per acre (after thinning) in the Lassen National Forest in California was trees.6 Let refer to the number of trees (after thinning) in the Lassen National Forest in California. Find the following probabilities:

  1. Probability that

6.3.23

(a) 0.0399 (b) 0.0360 (c) 0.0395

Question 6.206

24. Clickstream Analysis. “Clickstream analysis” refers to the analysis of the sequence of Web page clicks by users of a particular Web site. The mean number of page hits on the Web site of the Environmental Protection Agency (EPA) was page hits per visitor.7 Let refer to the number of page hits on the EPA Web site. Find the following probabilities:

  1. Probability that
  4. Probability that is less than 2
  5. Probability that is greater than 2. (Hint: Use the answers to (c) and (d).)

Question 6.207

25. DNA Mutation Rate. The mutation rate for human DNA is 0.0000000214 per base pair per generation.8 Suppose that we examine the 247 million base pairs in human chromosome 1 (the largest of the 23 human chromosomes). Do the following:

  1. Verify that the requirements are met for using the Poisson distribution to approximate the binomial distribution.
  2. Find the mean of the Poisson distribution used to approximate the binomial distribution.
  3. Approximate the probability that chromosome 1 will have mutations.

    348

  4. Approximate the probability that chromosome 1 will have at least one mutation.
  5. Approximate the probability that chromosome 1 will have at most two mutations.

6.3.25

(a) and (b) 5.2858 (c) 0.0051 (d) 0.9949 (e) 0.1026

Question 6.208

26. Due-Date Babies. Only 4% of babies are born on their due dates.9 Massachusetts General Hospital in Boston assists in the delivery of 142 babies per 2-week period.10 Do the following:

  1. Verify that the requirements are met for using the Poisson distribution to approximate the binomial distribution.
  2. Find the mean of the Poisson distribution used to approximate the binomial distribution.
  3. Approximate the probability that babies will be born on their due dates during a two-week period.
  4. Approximate the probability that at least one baby will be born on its due date during a two-week period.

Question 6.209

27. Red Sox Runs. In 2013, the Boston Red Sox led all Major League Baseball teams with a mean number of runs per game of (Source:www.teamrankings.com). Let refer to the number of runs per game for the Boston Red Sox. Find the following probabilities:

  1. Probability that
  3. Probability that the Red Sox get shut out (score zero runs)
  4. Probability that

6.3.27

(a) 0.1749 (b) 0.1684 (c) 0.0056 (d) 0

Question 6.210

28. Broward Burglaries. The mean number of burglaries taking place at the Central Campus of Broward Community College in Davie, Florida, is per year.11 Let refer to the number of burglaries taking place on that campus in a year. Find the following probabilities:

  1. Probability that
  2. Probability that is either 23 or 24
  3. Probability that no burglaries occur
  4. Probability that

Question 6.211

29. Football Fatalities. The mean number of fatalities from playing high school football in the United States is 3.8.12 Let refer to the number of fatalities from playing high school football in one year. Find the following probabilities:

  1. Probability that
  3. Probability that no fatalities occur in a year
  4. Probability that

6.3.29

(a) 0.2046 (b) 0.1615 (c) 0.0224 (d) 0.5265

Question 6.212

30. Social Media-Driven Holiday Sales. IBM reported that 1% of Black Friday purchases on ecommerce Web sites were driven directly by social media. Suppose that we take a sample of 200 such purchases. Do the following:

  1. Verify that the requirements are met for using the Poisson distribution to approximate the binomial distribution.
  2. Find the mean of the Poisson distribution used to approximate the binomial distribution.
  3. Approximate the probability that purchase was driven by social media.
  4. Approximate the probability that at least one purchase was driven by social media.

Question 6.213

31. Teens in Virtual Worlds. According to a Pew Research study, 10% of teenagers participate in online virtual worlds such as Second Life, Gaia, or Habbo Hotel.13 Suppose that we take a sample of 100 teenagers. Do the following:

  1. Verify that the requirements are met for using the Poisson distribution to approximate the binomial distribution.
  2. Find the mean of the Poisson distribution used to approximate the binomial distribution.
  3. Approximate the probability that teenagers participate in such online virtual worlds.
  4. Approximate the probability that either 10, or 11, or 12 teenagers participate in such online virtual worlds.

6.3.31

(a) and (b) 10 (c) 0.1251 (d) 0.3336

Question 6.214

32. Challenge Exercise. For the given exercise, identify the values of that would be considered (i) moderately unusual, and (ii) outliers.

  1. Exercise 27
  2. Exercise 28
  3. Exercise 29
  4. Exercise 30